Marches al\'eatoires dans un c\^one et fonctions discr\`etes harmoniques

TL;DR

This paper explores random walks in cones, highlighting their importance in probability, their connection to Brownian motion, and recent advances in constructing discrete harmonic functions.

Contribution

It provides an overview of key concepts, links to Brownian motion, and recent research developments in discrete harmonic functions related to random walks in cones.

Findings

Connection between random walks and Brownian motion in cones

Construction methods for discrete harmonic functions

Relevance to spectral theory and combinatorics

Abstract

Random walks in cones have the double interest of being at the heart of many probabilistic problems and of being related to many mathematical fields, such as spectral theory, combinatorics, or discrete complex analysis. In this article, we present some key ideas associated with these processes: we will discuss their definition, the link with Brownian motion in cones, as well as some recent research topics such as the construction of discrete harmonic functions.